Rotated shock tube

ID	Group	Milestone	Backends	Figure	Regression
MV-8	Ideal MHD	M2 (#41)	CPU, FNL	yes	rotated-shock-tube

Case and scope

Ryu–Jones 1a rotated in the $x$–$y$ plane (Tóth 2000, section 6.3.2) makes the dimension-by-dimension scheme solve a 1-D problem on truly 2-D data. The exact normal field stays uniform, $B_{\parallel }=5/\sqrt{4\pi }$, so its deviation measures the divergence error of the base scheme with GLM, and the whole solution must match the same problem solved in 1-D.

What the case catches:

• a divergence error of the multi-dimensional scheme: $\delta B_{\parallel }$ (Tóth eq. 45, against the analytic value) under a bound, decreasing with $N$;
• a 2-D solution that departs from the 1-D one: the mean relative $L_1$ error of $\rho$, $v_{\parallel }$, $v_{\perp }$, $p$, $B_{\parallel }$, $B_{\perp }$ against a 1-D reference run, under a bound, decreasing with $N$.

Models and numerics

Item	Choice	Code
Equations	ideal MHD, $\gamma =5/3$	models
Space	WENO5 flux splitting on characteristic variables	compute_residuals_weno, adam_flume_cpu_object.F90; mhd_glm_face_split_fluxes, adam_flume_mhd_library.F90
Divergence control	GLM, $c_h=24$ (the fastest signal, $\sim$20–24 in the ramp; glm_ch_check = error), glm_alpha = 0.18, glm_damping_length = 1.0	models
Floors	disabled	models
Time	SSP Runge–Kutta, 5 stages (runge-kutta-ssp-54), CFL 0.4	adam_rk_object.F90
Boundaries	periodic (FLUME has no shifted-periodic boundary)	boundary conditions
FNL backend	the same kernels on the device	adam_flume_fnl_object.F90

Initial and boundary conditions

Item	Value
Problem	Ryu & Jones (1995) 1a in the frame of the normal: $(\rho ,v_{\parallel },v_{\perp },p,B_{\parallel },B_{\perp })=(1,10,0,20,5/\sqrt{4\pi },5/\sqrt{4\pi })$ left, $(1,-10,0,1,5/\sqrt{4\pi },5/\sqrt{4\pi })$ right
Initial condition	rotated-riemann, $s=n_{x}x+n_{y}y$, period 1: the jump at $s=1/4+1/(4N)$ (off every cell centre), a linear ramp of the primitive variables back to the left state over $s\in (1/2,9/10]$
Normals	$(1,2)$, Tóth's ${\tan }^{-1}2=63.4°$; $(1,1)$, 45°
2-D grid	$[0,1{]}^2$, the vortex grid of V2 (quadtree, $4\times 4$ blocks), $N^2$ cells, $N=128,256$, periodic
1-D reference	the same problem along $x$ with normal $(|n|,0)$ on $[0,1/|n|]$, the Sod grid of V1 (4 blocks along $x$), $4\times 256=1024$ cells per period (4 times the normal resolution of $N=256$), periodic
Ranks	2
Final time	$t=0.04/|n|$: the fan of the jump (fast speeds $\sim 5$) stays within $0.2/|n|$ of it, as in Tóth ($t=0.08\cos a$ on his strip; fast shocks at 0.1 and 0.9 at $t=0.08$)

Input files

File	Role
src/tests/flume/verification/mhd/rotated-shock-tube/make_rotated_shock_tube.py	writes the 2-D input from vortex/vortex-n064.ini or the 1-D reference from sod/sod-x.ini (--angle 45|63, --cells, --reference, --cfl, --glm-ch)
src/tests/flume/verification/mhd/rotated-shock-tube/check.sh	runs the reference and the ladder for both normals (--np N, --legs LIST: mv8-63, mv8-45; about 30 min)
src/tests/flume/verification/make_doc_runs.sh <runs> rotated, make_doc_figures.py --only rotated-shock-tube	re-runs both normals at $N=256$ and their 1-D references with checkpoints kept (4 runs, about 40 min)
src/tests/flume/verification/mhd/rotated-shock-tube/rotated_shock_tube_oracle.py	$\delta B_{\parallel }$ and the mean relative error against the reference

cd src/tests/flume/verification/mhd
rotated-shock-tube/check.sh                                                     # CPU
FLUME_EXE=$PWD/../../../../../exe/adam_flume_fnl rotated-shock-tube/check.sh    # FNL

Results

The ${256}^2$ runs at 63.4° (top) and 45° (bottom), CPU. Left: $B_{\parallel }/B_0-1$, the deviation of the normal field from its exact uniform value, at most 1.5 % (63.4°) and 2.5 % (45°), on the discontinuities. Centre and right: the density and the tangential field of every cell against $s=\mathbf{n}\cdot \mathbf{x}$, over the 1-D reference; the 2-D runs smear the peaks of the fan the reference resolves, the source of the mean error in the table.

Check	Bound (CPU baseline M2-P6d + 2%)	CPU	FNL
$\delta B_{\parallel }$, 63.4°, $N=128/256$	$6.146\cdot {10}^{-3}$ / $2.842\cdot {10}^{-3}$	below / $2.79\cdot {10}^{-3}$	pass
mean relative error, 63.4°, $N=128/256$	$8.989\cdot {10}^{-2}$ / $4.095\cdot {10}^{-2}$	below	pass
$\delta B_{\parallel }$, 45°, $N=128/256$	$6.424\cdot {10}^{-3}$ / $4.252\cdot {10}^{-3}$	below / $4.17\cdot {10}^{-3}$	pass
mean relative error, 45°, $N=128/256$	$5.440\cdot {10}^{-2}$ / $2.600\cdot {10}^{-2}$	below	pass
both measures decreasing with $N$	strict	pass	pass

Reference values of Tóth's second-order base scheme at 63.4°, $N=256$ on his strip (twice the cells across the fan of $N=256$ here): $\delta B_{\parallel }=3.7\cdot {10}^{-3}$, mean error 0.0238.

FNL ran both normals (work directories of both backends); the tables record the CPU values.

Related cases: Ryu–Jones 2a is the axis-aligned MHD Riemann problem; field loop measures the divergence error of a smooth field; GLM cleaning the cleaning itself.

Findings

• $\delta B_{\parallel }$ is below Tóth's base-scheme value at 63.4° ($2.79\cdot {10}^{-3}$ against $3.7\cdot {10}^{-3}$) with half his cells across the fan.
• The 45° normal is not degenerate for this scheme, unlike Tóth's CT and CD schemes: $B_{\parallel }$ is conserved to truncation, not to round-off.
• The convergence of $\delta B_{\parallel }$ is about first order at 63.4° and slower at 45° (rates about 1.1 and 0.6 from 128 to 256, computed from the baselines behind the bounds).
• A periodic line forces the velocity jump to be undone. A second jump would be the mirror problem, an expansion close to vacuum (density $\sim 2\cdot {10}^{-4}$, Alfvén speed $\sim 100$) where $max(|u|+c_f)$ grows with the resolution of the fan and no $c_h$ holds; the resolved ramp (density $\sim 1/3$ at the final time) avoids it, and the 1-D reference solves the same problem, ramp and wave interactions included.